Problem: Find the product of the least common multiple (LCM) of $8$ and $6$ and the greatest common divisor (GCD) of $8$ and $6$.
Since $6 = 2 \cdot 3$ and $3$ is relatively prime with $8$ while $2$ divides into both $6$ and $8$, it follows that $\text{gcd}(6,8) = 2$. The multiples of $8$ are $8, 16, 24, 32\ldots$ and the multiples of $6$ are $6, 12, 18, 24, 30, \ldots$, so $\text{lcm}(6,8) = 24$. Thus, the desired product is $2 \times 24 = \boxed{48}.$

Notice that this product is equal to the product of the original numbers, $6\times 8$. Is there a reason for that?